Stefan problem for a one-dimensional heat-conduction hyperbolic system (Q942009)

The main purpose of the paper is to build the Riemann function of order two, for a 1D heat-conduction hyperbolic system of two equations. The author considers the hyperbolic operator \(L=\partial /\partial t+A(x)\partial /\partial s+B(x)\), where \(A\) is a \(2\times 2\) diagonal matrix with \(C^{1}\) functions and \(B\) is a \(2\times 2\) matrix with \(C^{0}\) functions. The author defines the \(2\times 2\) Riemann matrix \(V\) for this system through the solution of some Goursat problem \(L(V_{0})=0\), with appropriate boundary conditions along the characteristics which involve two Riemann functions \( U_{1}\) and \(U_{2}\) of the first kind. A first result of the paper proves that if a Cauchy problem associated to \(L\) is solvable in a class \(S_{L}\) of smooth functions, the solution can be represented in terms of \(U_{1}\), \(U_{2} \) and \(V\). Then the author specializes to a Stefan problem involving \(L\) for an appropriate structure of the matrix \(B\). She here computes the Riemann matrix \(V\) as a series of vectors defined in a recursive way. The main result of the paper proves the solvability of this Stefan problem under some further hypotheses on the data. This extends in an non-homogeneous case earlier results.